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The usual unary and binary ring operations are available
for noncommutative polynomials, noting that multiplication is associative
but noncommutative, of course.
+ a : AlgFrElt -> AlgFrElt
- a : AlgFrElt -> AlgFrElt
a + b : AlgFrElt, AlgFrElt -> AlgFrElt
a - b : AlgFrElt, AlgFrElt -> AlgFrElt
a * b : AlgFrElt, AlgFrElt -> AlgFrElt
a ^ k : AlgFrElt, RngIntElt -> AlgFrElt
a / b : AlgFrElt, AlgFrElt -> FldFunMElt
a div b : AlgFrElt, AlgFrElt -> AlgFrElt
a +:= b : AlgFrElt, AlgFrElt -> AlgFrElt
a -:= b : AlgFrElt, AlgFrElt -> AlgFrElt
a *:= b : AlgFrElt, AlgFrElt -> AlgFrElt
a div:= b : AlgFrElt, AlgFrElt -> AlgFrElt
a eq b : AlgFrElt, AlgFrElt -> BoolElt
a ne b : AlgFrElt, AlgFrElt -> BoolElt
a in R : AlgFrElt, Rng -> BoolElt
a notin R : AlgFrElt, Rng -> BoolElt
IsZero(f) : AlgFrElt -> BoolElt
IsOne(f) : AlgFrElt -> BoolElt
IsMinusOne(f) : AlgFrElt -> BoolElt
IsNilpotent(f) : AlgFrElt -> BoolElt
IsIdempotent(f) : AlgFrElt -> BoolElt
IsUnit(f) : AlgFrElt -> BoolElt
IsZeroDivisor(f) : AlgFrElt -> BoolElt
IsRegular(f) : AlgFrElt -> BoolElt
IsIrreducible(f) : AlgFrElt -> BoolElt
IsPrime(f) : AlgFrElt -> BoolElt
The functions in this subsection allow one to access noncommutative
polynomials.
Given a noncommutative polynomial f with coefficients in R, this
function returns a sequence of `base' coefficients, that is, a
sequence of elements of R occurring as
coefficients of the monomials in f. Note that the monomials
are ordered, and that the sequence of coefficients corresponds
exactly to the sequence of monomials returned by Monomials(f).
Given a noncommutative polynomial f with coefficients in R, this
function returns the leading coefficient of f as an element
of R; this is the coefficient of the leading monomial of f,
that is, the first among the monomials occurring
in f with respect to the ordering of monomials used in F.
Given a noncommutative polynomial f with coefficients in R, this
function returns the trailing coefficient of f as an element
of R; this is the coefficient of the trailing monomial of f,
that is, the last among the monomials occurring
in f with respect to the ordering of monomials used in F.
Given a noncommutative polynomial f and a monomial m, this function
returns the coefficient with which m occurs in f as an element of R.
Given a noncommutative polynomial f∈F, this function returns a
sequence of the monomials (monoid words) occurring in f. Note that
the monomials in F are ordered, and that the sequence of monomials
corresponds exactly to the sequence of coefficients returned by Coefficients(f).
Given a noncommutative polynomial f∈F this function returns
the leading monomial of f, that is, the first monomial element
of F that occurs in f, with respect to the
ordering of monomials used in F.
Given a noncommutative polynomial f∈F, this function returns the sequence
of (non-zero) terms of f as elements of F. The terms are ordered
according to the ordering on the monomials in F. Consequently
the i-th element of this sequence of terms will be equal to
the product of the i-th element of the sequence of coefficients
and the i-th element of the sequence of monomials.
Given a noncommutative polynomial f∈F, this
function returns the leading term of f as an element
of F; this is the product of the leading monomial and the leading
coefficient
that is, the first among the monomial terms occurring
in f with respect to the ordering of monomials used in F.
Given a noncommutative polynomial f∈F, this
function returns the trailing term of f as an element
of F; this is
the last among the monomial terms occurring
in f with respect to the ordering of monomials used in F.
Given a noncommutative monomial (word) m, return the length of m, i.e.,
the number of letters of m. Note that this differs from the commutative
case, where the number of terms in a polynomial is returned.
Given a noncommutative monomial (word) m of length l, and an integer
i with 1≤i≤l, return the i-th letter of m.
Given a noncommutative polynomial f, this
function returns the total degree of f, which is the maximum
of the lengths of all monomials that occur in f.
If f is the zero polynomial, the return value is -1.
Given a noncommutative polynomial, this
function returns the leading total degree of f, which is the
length of the leading monomial of f.
In this example we illustrate the above access functions.
> K := RationalField();
> F<x,y,z> := FreeAlgebra(K, 3);
> f := (3*x*y - 2*y*z)*(4*x - 7*z*y) + 23*x*y*z;
> f;
-21*x*y*z*y + 14*y*z^2*y + 12*x*y*x + 23*x*y*z - 8*y*z*x
> TotalDegree(f);
4
> Coefficients(f);
[ -21, 14, 12, 23, -8 ]
> Monomials(f);
[
x*y*z*y,
y*z^2*y,
x*y*x,
x*y*z,
y*z*x
]
> Terms(f);
[
-21*x*y*z*y,
14*y*z^2*y,
12*x*y*x,
23*x*y*z,
-8*y*z*x
]
> MonomialCoefficient(f, x*y*z);
23
> LeadingTerm(f);
-21*x*y*z*y
> LeadingCoefficient(f);
-21
> m := Monomials(f)[1];
> m;
x*y*z*y
> Length(m);
4
> m[1];
x
> m[2];
y
Evaluate(f, s) : AlgFrElt, < RngElt, ..., RngElt > -> RngElt
Given an element f of a free algebra F=R< x1, ..., xn >
and a sequence or tuple s of ring or algebra elements of length n,
return the value of f at s, that is, the value obtained by
substituting xi=s[i]. This behaves in the same way as the hom
constructor above.
If the elements of
s lie in a ring and can be lifted into the coefficient ring R,
then the result will be an element of R.
If the elements of
s cannot be lifted to the coefficient ring, then an attempt is made
to do a generic evaluation of f at s. In this case, the result will
be of the same type as the elements of s.
In this example we illustrate the above access functions.
> K := RationalField();
> F<x,y,z> := FreeAlgebra(K, 3);
> g := x*y + y*z;
> g;
x*y + y*z
> Evaluate(g, [1,2,3]);
8
> Parent($1);
Rational Field
> Evaluate(g, [y,x,z]);
x*z + y*x
> Parent($1);
Finitely presented algebra of rank 3 over Rational Field
Non-commutative Graded Lexicographical Order
Variables: x, y, z
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